Lesson 17: Vector AutoRegressive Models
|
|
- Posy Butler
- 6 years ago
- Views:
Transcription
1 Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@ec.univaq.it
2 Vector AutoRegressive models The extension of ARMA models into a multivariate framework leads to Vector AutoRegressive (VAR) models Vector Moving Average (VMA) models Vector AutoregRegressive Moving Average (VARMA) models
3 Vector AutoRegressive models The Vector AutoRegressive (VAR) models, made famous in Chris Sims s paper Macroeconomics and Reality, Econometrica, 1980, are one of the most applied models in the empirical economics.
4 Vector AutoRegressive models Let be a K-variate random process. { yt = (y 1t,..., y Kt ) ; t Z }
5 Vector AutoRegressive models We say that the process {y t ; t Z} follows a vector autoregressive model of order p, denoted VAR(p) if y t = ν + A 1 y t A p y t p + u t, t Z p is a positive integer, A i are fixed (K K) coefficient matrices, ν = (ν 1,..., ν K ) is a fixed (K 1) vector of intercept terms, u t = (u 1t,..., u Kt ) is ak-dimensional white noise with covariance matrix Σ u. The covariance matrix Σ u is assumed to be nonsingular.
6 Example: Bivariate VAR(1) or [ y1t y 2t ] = [ ν1 ν 2 ] [ a11 a + 12 a 21 a 22 ] [ y1t 1 y 2t 1 y 1t = ν 1 + a 11 y 1t 1 + a 12 y 2t 1 + u 1t ] [ u1t + u 2t ] y 2t = ν 1 + a 21 y 1t 1 + a 22 y 2t 1 + u 2t where cov(u 1t, u 2s ) = σ 12 for t = s, 0 otherwise
7 Advantages of VAR models 1 They are easy to estimate. 2 They have good forecasting capabilities. 3 The researcher does not need to specify which variables are endogenous or exogeneous. All are endogenous. 4 In a VAR system is very easy to test for Granger non-causality.
8 Problems with VARs 1 So many parameters. If there are K equations, one for each K variables and p lags of each of the variables in each equation, (K + pk 2 ) parameters will have to be estimated. 2 VARs are a-theoretical, since they use little economic theory. Thus VARs cannot used to obtain economic policy prescriptions.
9 Estimation In this lesson, the estimation of a vector autoregressive model is discussed.
10 Estimation Consider the VAR(1) y 1t = ν 1 + a 11 y 1t 1 + a 12 y 2t 1 + u 1t y 2t = ν 1 + a 21 y 1t 1 + a 22 y 2t 1 + u 2t where cov(u 1t, u 2s ) = σ 12 for t = s, 0 otherwise.
11 Estimation The model corresponds to 2 regressions with different dependent variables and identical explanatory variables. We could estimate this model using the ordinary least squares (OLS) estimator computed separately from each equations.
12 Estimation It is assumed that a time series of the y variables is available. In addition, a presample value is assumed to be available. y 1 = [y 11, y 21 ],..., y T = [y 1T, y 2T ] y 0 = [y 10, y 20 ]
13 Estimation Consider the first equation y 1t = ν 1 + a 11 y 10 + a 12 y 2t 1 + u 1t ; t = 1,..., T y 11 = ν 1 + a 11 y 10 + a 12 y 20 + u 11 y 12 = ν 1 + a 11 y 11 + a 12 y 21 + u 12. y 1T = ν 1 + a 11 y 1T 1 + a 12 y 2T 1 + u 1T
14 Estimation We define y 1 = [y 11,..., y 1T ], 1 y 1,0 y 2,0 1 y 1,1 y 2,1 X 1 =... 1 y 1,T 1 y 2,T 1, π 1 = [ν 1, a 11, a 12 ], u 1 = [u 11,..., u 1T ],
15 Thus y 1 = Xπ 1 + u, The OLS estimator π 1 is given by ˆπ 1 = (X X) 1 X y 1
16 Estimation Consider the second equation y 2t = ν 2 + a 21 y 10 + a 22 y 2t 1 + u 2t ; t = 1,..., T y 21 = ν 2 + a 21 y 10 + a 22 y 20 + u 21 y 22 = ν 2 + a 21 y 11 + a 22 y 21 + u 22. y 2T = ν 2 + a 21 y 1T 1 + a 22 y 2T 1 + u 2T
17 Estimation We define y 2 = [y 21,..., y 2T ], π 2 = [ν 2, a 21, a 22 ], u 2 = [u 21,..., u 2T ],
18 Estimation Thus y 2 = Xπ 2 + u 2, The OLS estimator π 2 is given by ˆπ 2 = (X X) 1 X y 2
19 OLS estimators ˆπ 1 = (X X) 1 X y 1 π 1 = [ν 1, a 11, a 12 ] ˆπ 2 = (X X) 1 X y 2 π 2 = [ν 2, a 21, a 22 ]
20 OLS Estimation Are the OLS estimators for π 2 and π 2 efficient estimators?
21 Seemingly Unrelated Regressions Estimation We remember that cov(u 1t, u 2t ) = σ 12 0 When contemporaneous correlation exists, it may be more efficient to estimate all equations jointly, rather than to estimate each one separately using least squares. The appropriate joint estimation technique is the GLS estimation
22 Seemingly Unrelated Regressions Equations Let as consider the following set of two equations y 1t = β 10 + β 11 x 1t + β 12 z 1t + u 1t y 2t = β 10 + β 11 x 2t + β 12 z 2t + u 2t
23 Seemingly Unrelated Regressions Equations The two equation can be written in the usual matrix algebra notation as y 1 = X 1 β 1 + u 1 y 2 = X 2 β 2 + u 2
24 Seemingly Unrelated Regressions Equations where y i = [y i1,..., y it ] i = 1, 2 1 x i,1 z i,1 1 x i,2 z i,2 X i =... 1 x i,t z i,t i = 1, 2, β i = [β i0, β i1, β i2 ], i = 1, 2, and u i = [u i1,..., u it ], i = 1, 2,
25 Seemingly Unrelated Regressions Equations Further, we assume that 1 E[u it ] = 0 i = 1, 2 t = 1, 2...T 2 var(u it ) = σi 2 i = 1, 2 t = 1, 2...T ; 3 E[u it u jt ] = σ ij i, j = 1, 2; 4 E[u it u js ] = 0 for t s and i, j = 1, 2;
26 Seemingly Unrelated Regressions Equations Now we write the two equations as the folllowing super model [ ] [ ] [ ] [ ] y1 X1 0 β1 u1 = + 0 X 2 β 2 u 2 or y 2 y = Xβ + u The first point to note is that it can be shown that least squares applied to this system is identical to applying least squares separately to each of the two equations.
27 Seemingly Unrelated Regressions Equations Note that the covariance matrix of the joint disturbace vector u is given by [ ] [ ] σ 2 Φ = 1 I σ 12 I σ 2 σ 12 I σ2 2I = 1 σ 12 σ 12 σ2 2 I T = Σ I T
28 Seemingly Unrelated Regressions Equations The disturbance covariance matrix Φ is of dimension (2T 2T ).An important point to note is that Φ cannot be written as scalar multiplied by a 2T -dimensional identity matrix. Thus the best linear unbiased estimator for β is given by the generalized least squares estimator ˆβ GLS = (X Φ 1 X) 1 X Φ 1 y = [X (Σ 1 I T )X] 1 X(Σ 1 I T )y It has lower variance than the least squares estimators for β because it takes into account the contemporaneous correlation between the disturbances in different equation.
29 Seemingly Unrelated Regressions Equations There are two conditions under the which least squares is identical to generalized least squares. 1 The first is when all contemporaneus covariances are zero, σ 12 = 0. 2 The second is when the explanatory variables in each equation are identical, X 1 = X 2
30 Estimation of A VAR model The use of generalized least squares estimator does not lead to a gain in efficiency when each equation contains the same explanatory variables. Thus the autoregressive cooefficients of our VAR(1) model can be estimated, without loss of estimation efficiency, by ordinary least squares. This in turn is equivalent to estimating each equation separately by OLS
31 Estimation of A VAR model The (2 2) unknown covariance matrix Σ may be consistent estimated by ˆΣ whose elements are ˆσ ij = û iûj T 2p 1 for i, j = 1, 2 where û i = y i Xˆπ i
32 Estimation of A VAR model In general, the autoregressive cooefficients of a K-dimensional VAR(p) model y t = ν + A 1 y t A p y t p + u t can be estimated, without loss of estimation efficiency, by ordinary least squares.in the first equation, we have to run the regression y 1t on y 1t 1,..., y Kt 1,..., y 1t p,..., y Kt p, in the second equation, we regress and so on. y 12 on y 1t 1,..., y Kt 1,..., y 1t p,..., y Kt p,
33 Conclusion Since the VAR(p) model is just a Seemingly Unrelated Regression (SUR) model where each equation has the same explanatory variables, each equation may be estimated separately by ordinary least squares without losing efficiency relative to generalized least squares.
Multivariate forecasting with VAR models
Multivariate forecasting with VAR models Franz Eigner University of Vienna UK Econometric Forecasting Prof. Robert Kunst 16th June 2009 Overview Vector autoregressive model univariate forecasting multivariate
More informationF9 F10: Autocorrelation
F9 F10: Autocorrelation Feng Li Department of Statistics, Stockholm University Introduction In the classic regression model we assume cov(u i, u j x i, x k ) = E(u i, u j ) = 0 What if we break the assumption?
More informationEconometría 2: Análisis de series de Tiempo
Econometría 2: Análisis de series de Tiempo Karoll GOMEZ kgomezp@unal.edu.co http://karollgomez.wordpress.com Segundo semestre 2016 IX. Vector Time Series Models VARMA Models A. 1. Motivation: The vector
More informationLesson 4: Stationary stochastic processes
Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@univaq.it Stationary stochastic processes Stationarity is a rather intuitive concept, it means
More informationSimultaneous Equation Models Learning Objectives Introduction Introduction (2) Introduction (3) Solving the Model structural equations
Simultaneous Equation Models. Introduction: basic definitions 2. Consequences of ignoring simultaneity 3. The identification problem 4. Estimation of simultaneous equation models 5. Example: IS LM model
More informationTitle. Description. var intro Introduction to vector autoregressive models
Title var intro Introduction to vector autoregressive models Description Stata has a suite of commands for fitting, forecasting, interpreting, and performing inference on vector autoregressive (VAR) models
More informationLesson 2: What is a time series Model
Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@ec.univaq.it Time series forecasts Having observed a time series {x 1, x 2,..., x T } over the
More informationFinancial Econometrics
Financial Econometrics Multivariate Time Series Analysis: VAR Gerald P. Dwyer Trinity College, Dublin January 2013 GPD (TCD) VAR 01/13 1 / 25 Structural equations Suppose have simultaneous system for supply
More informationLesson 9: Autoregressive-Moving Average (ARMA) models
Lesson 9: Autoregressive-Moving Average (ARMA) models Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@ec.univaq.it Introduction We have seen
More informationMultivariate Time Series Analysis and Its Applications [Tsay (2005), chapter 8]
1 Multivariate Time Series Analysis and Its Applications [Tsay (2005), chapter 8] Insights: Price movements in one market can spread easily and instantly to another market [economic globalization and internet
More informationLinear Regression. Junhui Qian. October 27, 2014
Linear Regression Junhui Qian October 27, 2014 Outline The Model Estimation Ordinary Least Square Method of Moments Maximum Likelihood Estimation Properties of OLS Estimator Unbiasedness Consistency Efficiency
More informationMultivariate Models. Christopher Ting. Christopher Ting. April 19,
Multivariate Models Chapter 7 of Chris Brook s Book Christopher Ting http://www.mysmu.edu/faculty/christophert/ Christopher Ting : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 April 19, 2017 Christopher
More informationPhD/MA Econometrics Examination. January, 2015 PART A. (Answer any TWO from Part A)
PhD/MA Econometrics Examination January, 2015 Total Time: 8 hours MA students are required to answer from A and B. PhD students are required to answer from A, B, and C. PART A (Answer any TWO from Part
More informationVector Autoregression
Vector Autoregression Jamie Monogan University of Georgia February 27, 2018 Jamie Monogan (UGA) Vector Autoregression February 27, 2018 1 / 17 Objectives By the end of these meetings, participants should
More informationA Non-Parametric Approach of Heteroskedasticity Robust Estimation of Vector-Autoregressive (VAR) Models
Journal of Finance and Investment Analysis, vol.1, no.1, 2012, 55-67 ISSN: 2241-0988 (print version), 2241-0996 (online) International Scientific Press, 2012 A Non-Parametric Approach of Heteroskedasticity
More informationVector Auto-Regressive Models
Vector Auto-Regressive Models Laurent Ferrara 1 1 University of Paris Nanterre M2 Oct. 2018 Overview of the presentation 1. Vector Auto-Regressions Definition Estimation Testing 2. Impulse responses functions
More informationVAR Models and Applications
VAR Models and Applications Laurent Ferrara 1 1 University of Paris West M2 EIPMC Oct. 2016 Overview of the presentation 1. Vector Auto-Regressions Definition Estimation Testing 2. Impulse responses functions
More informationINTRODUCTORY ECONOMETRICS
INTRODUCTORY ECONOMETRICS Lesson 2b Dr Javier Fernández etpfemaj@ehu.es Dpt. of Econometrics & Statistics UPV EHU c J Fernández (EA3-UPV/EHU), February 21, 2009 Introductory Econometrics - p. 1/192 GLRM:
More informationVAR Model. (k-variate) VAR(p) model (in the Reduced Form): Y t-2. Y t-1 = A + B 1. Y t + B 2. Y t-p. + ε t. + + B p. where:
VAR Model (k-variate VAR(p model (in the Reduced Form: where: Y t = A + B 1 Y t-1 + B 2 Y t-2 + + B p Y t-p + ε t Y t = (y 1t, y 2t,, y kt : a (k x 1 vector of time series variables A: a (k x 1 vector
More informationTwo-Variable Regression Model: The Problem of Estimation
Two-Variable Regression Model: The Problem of Estimation Introducing the Ordinary Least Squares Estimator Jamie Monogan University of Georgia Intermediate Political Methodology Jamie Monogan (UGA) Two-Variable
More informationPanel Data Models. James L. Powell Department of Economics University of California, Berkeley
Panel Data Models James L. Powell Department of Economics University of California, Berkeley Overview Like Zellner s seemingly unrelated regression models, the dependent and explanatory variables for panel
More informationLecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16)
Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16) 1 2 Model Consider a system of two regressions y 1 = β 1 y 2 + u 1 (1) y 2 = β 2 y 1 + u 2 (2) This is a simultaneous equation model
More informationEcon 423 Lecture Notes: Additional Topics in Time Series 1
Econ 423 Lecture Notes: Additional Topics in Time Series 1 John C. Chao April 25, 2017 1 These notes are based in large part on Chapter 16 of Stock and Watson (2011). They are for instructional purposes
More informationLesson 7: Estimation of Autocorrelation and Partial Autocorrela
Lesson 7: Estimation of Autocorrelation and Partial Autocorrelation Function Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@ec.univaq.it Estimation
More informationECON 4160, Spring term Lecture 12
ECON 4160, Spring term 2013. Lecture 12 Non-stationarity and co-integration 2/2 Ragnar Nymoen Department of Economics 13 Nov 2013 1 / 53 Introduction I So far we have considered: Stationary VAR, with deterministic
More informationFinal Exam. Economics 835: Econometrics. Fall 2010
Final Exam Economics 835: Econometrics Fall 2010 Please answer the question I ask - no more and no less - and remember that the correct answer is often short and simple. 1 Some short questions a) For each
More informationEconometric Methods. Prediction / Violation of A-Assumptions. Burcu Erdogan. Universität Trier WS 2011/2012
Econometric Methods Prediction / Violation of A-Assumptions Burcu Erdogan Universität Trier WS 2011/2012 (Universität Trier) Econometric Methods 30.11.2011 1 / 42 Moving on to... 1 Prediction 2 Violation
More informationEconometrics I KS. Module 2: Multivariate Linear Regression. Alexander Ahammer. This version: April 16, 2018
Econometrics I KS Module 2: Multivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: April 16, 2018 Alexander Ahammer (JKU) Module 2: Multivariate
More informationECON 4160, Lecture 11 and 12
ECON 4160, 2016. Lecture 11 and 12 Co-integration Ragnar Nymoen Department of Economics 9 November 2017 1 / 43 Introduction I So far we have considered: Stationary VAR ( no unit roots ) Standard inference
More informationA primer on Structural VARs
A primer on Structural VARs Claudia Foroni Norges Bank 10 November 2014 Structural VARs 1/ 26 Refresh: what is a VAR? VAR (p) : where y t K 1 y t = ν + B 1 y t 1 +... + B p y t p + u t, (1) = ( y 1t...
More informationGLS and FGLS. Econ 671. Purdue University. Justin L. Tobias (Purdue) GLS and FGLS 1 / 22
GLS and FGLS Econ 671 Purdue University Justin L. Tobias (Purdue) GLS and FGLS 1 / 22 In this lecture we continue to discuss properties associated with the GLS estimator. In addition we discuss the practical
More informationEconometrics Summary Algebraic and Statistical Preliminaries
Econometrics Summary Algebraic and Statistical Preliminaries Elasticity: The point elasticity of Y with respect to L is given by α = ( Y/ L)/(Y/L). The arc elasticity is given by ( Y/ L)/(Y/L), when L
More informationVector Autoregressive Model. Vector Autoregressions II. Estimation of Vector Autoregressions II. Estimation of Vector Autoregressions I.
Vector Autoregressive Model Vector Autoregressions II Empirical Macroeconomics - Lect 2 Dr. Ana Beatriz Galvao Queen Mary University of London January 2012 A VAR(p) model of the m 1 vector of time series
More informationLesson 15: Building ARMA models. Examples
Lesson 15: Building ARMA models. Examples Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@ec.univaq.it Examples In this lesson, in order to illustrate
More informationEconometrics. Week 4. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 4 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 23 Recommended Reading For the today Serial correlation and heteroskedasticity in
More informationEconometrics I. Professor William Greene Stern School of Business Department of Economics 25-1/25. Part 25: Time Series
Econometrics I Professor William Greene Stern School of Business Department of Economics 25-1/25 Econometrics I Part 25 Time Series 25-2/25 Modeling an Economic Time Series Observed y 0, y 1,, y t, What
More informationLesson 8: Testing for IID Hypothesis with the correlogram
Lesson 8: Testing for IID Hypothesis with the correlogram Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@ec.univaq.it Testing for i.i.d. Hypothesis
More informationNotes on Time Series Modeling
Notes on Time Series Modeling Garey Ramey University of California, San Diego January 17 1 Stationary processes De nition A stochastic process is any set of random variables y t indexed by t T : fy t g
More informationThe Simple Regression Model. Part II. The Simple Regression Model
Part II The Simple Regression Model As of Sep 22, 2015 Definition 1 The Simple Regression Model Definition Estimation of the model, OLS OLS Statistics Algebraic properties Goodness-of-Fit, the R-square
More informationThe regression model with one stochastic regressor (part II)
The regression model with one stochastic regressor (part II) 3150/4150 Lecture 7 Ragnar Nymoen 6 Feb 2012 We will finish Lecture topic 4: The regression model with stochastic regressor We will first look
More informationEconometrics Multiple Regression Analysis: Heteroskedasticity
Econometrics Multiple Regression Analysis: João Valle e Azevedo Faculdade de Economia Universidade Nova de Lisboa Spring Semester João Valle e Azevedo (FEUNL) Econometrics Lisbon, April 2011 1 / 19 Properties
More informationSeemingly Unrelated Regressions in Panel Models. Presented by Catherine Keppel, Michael Anreiter and Michael Greinecker
Seemingly Unrelated Regressions in Panel Models Presented by Catherine Keppel, Michael Anreiter and Michael Greinecker Arnold Zellner An Efficient Method of Estimating Seemingly Unrelated Regressions and
More informationStatistics 910, #5 1. Regression Methods
Statistics 910, #5 1 Overview Regression Methods 1. Idea: effects of dependence 2. Examples of estimation (in R) 3. Review of regression 4. Comparisons and relative efficiencies Idea Decomposition Well-known
More informationProblem Set #6: OLS. Economics 835: Econometrics. Fall 2012
Problem Set #6: OLS Economics 835: Econometrics Fall 202 A preliminary result Suppose we have a random sample of size n on the scalar random variables (x, y) with finite means, variances, and covariance.
More informationEconometrics. Week 8. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 8 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 25 Recommended Reading For the today Instrumental Variables Estimation and Two Stage
More informationECON The Simple Regression Model
ECON 351 - The Simple Regression Model Maggie Jones 1 / 41 The Simple Regression Model Our starting point will be the simple regression model where we look at the relationship between two variables In
More informationECON3150/4150 Spring 2015
ECON3150/4150 Spring 2015 Lecture 3&4 - The linear regression model Siv-Elisabeth Skjelbred University of Oslo January 29, 2015 1 / 67 Chapter 4 in S&W Section 17.1 in S&W (extended OLS assumptions) 2
More informationVector Time-Series Models
T. Rothenberg Fall, 2007 1 Introduction Vector Time-Series Models The n-dimensional, mean-zero vector process fz t g is said to be weakly stationary if secondorder moments exist and only depend on time
More informationThe Statistical Property of Ordinary Least Squares
The Statistical Property of Ordinary Least Squares The linear equation, on which we apply the OLS is y t = X t β + u t Then, as we have derived, the OLS estimator is ˆβ = [ X T X] 1 X T y Then, substituting
More informationHeteroskedasticity and Autocorrelation
Lesson 7 Heteroskedasticity and Autocorrelation Pilar González and Susan Orbe Dpt. Applied Economics III (Econometrics and Statistics) Pilar González and Susan Orbe OCW 2014 Lesson 7. Heteroskedasticity
More informationLecture 1: OLS derivations and inference
Lecture 1: OLS derivations and inference Econometric Methods Warsaw School of Economics (1) OLS 1 / 43 Outline 1 Introduction Course information Econometrics: a reminder Preliminary data exploration 2
More informationTIME SERIES DATA ANALYSIS USING EVIEWS
TIME SERIES DATA ANALYSIS USING EVIEWS I Gusti Ngurah Agung Graduate School Of Management Faculty Of Economics University Of Indonesia Ph.D. in Biostatistics and MSc. in Mathematical Statistics from University
More informationIntroductory Econometrics
Based on the textbook by Wooldridge: : A Modern Approach Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna December 17, 2012 Outline Heteroskedasticity
More informationEconometrics Master in Business and Quantitative Methods
Econometrics Master in Business and Quantitative Methods Helena Veiga Universidad Carlos III de Madrid Models with discrete dependent variables and applications of panel data methods in all fields of economics
More informationVector autoregressions, VAR
1 / 45 Vector autoregressions, VAR Chapter 2 Financial Econometrics Michael Hauser WS17/18 2 / 45 Content Cross-correlations VAR model in standard/reduced form Properties of VAR(1), VAR(p) Structural VAR,
More informationModel Mis-specification
Model Mis-specification Carlo Favero Favero () Model Mis-specification 1 / 28 Model Mis-specification Each specification can be interpreted of the result of a reduction process, what happens if the reduction
More informationEconometrics II. Seppo Pynnönen. Spring Department of Mathematics and Statistics, University of Vaasa, Finland
Department of Mathematics and Statistics, University of Vaasa, Finland Spring 218 Part VI Vector Autoregression As of Feb 21, 218 1 Vector Autoregression (VAR) Background The Model Defining the order of
More informationDynamic Regression Models (Lect 15)
Dynamic Regression Models (Lect 15) Ragnar Nymoen University of Oslo 21 March 2013 1 / 17 HGL: Ch 9; BN: Kap 10 The HGL Ch 9 is a long chapter, and the testing for autocorrelation part we have already
More informationSection 2 NABE ASTEF 65
Section 2 NABE ASTEF 65 Econometric (Structural) Models 66 67 The Multiple Regression Model 68 69 Assumptions 70 Components of Model Endogenous variables -- Dependent variables, values of which are determined
More informationLECTURE 2 LINEAR REGRESSION MODEL AND OLS
SEPTEMBER 29, 2014 LECTURE 2 LINEAR REGRESSION MODEL AND OLS Definitions A common question in econometrics is to study the effect of one group of variables X i, usually called the regressors, on another
More informationStructural VAR Models and Applications
Structural VAR Models and Applications Laurent Ferrara 1 1 University of Paris Nanterre M2 Oct. 2018 SVAR: Objectives Whereas the VAR model is able to capture efficiently the interactions between the different
More informationRegression with time series
Regression with time series Class Notes Manuel Arellano February 22, 2018 1 Classical regression model with time series Model and assumptions The basic assumption is E y t x 1,, x T = E y t x t = x tβ
More informationS-GSTAR-SUR Model for Seasonal Spatio Temporal Data Forecasting ABSTRACT
Malaysian Journal of Mathematical Sciences (S) March : 53-65 (26) Special Issue: The th IMT-GT International Conference on Mathematics, Statistics and its Applications 24 (ICMSA 24) MALAYSIAN JOURNAL OF
More informationLinear models. Linear models are computationally convenient and remain widely used in. applied econometric research
Linear models Linear models are computationally convenient and remain widely used in applied econometric research Our main focus in these lectures will be on single equation linear models of the form y
More informationECON 4160: Econometrics-Modelling and Systems Estimation Lecture 9: Multiple equation models II
ECON 4160: Econometrics-Modelling and Systems Estimation Lecture 9: Multiple equation models II Ragnar Nymoen Department of Economics University of Oslo 9 October 2018 The reference to this lecture is:
More informationChapter 2: simple regression model
Chapter 2: simple regression model Goal: understand how to estimate and more importantly interpret the simple regression Reading: chapter 2 of the textbook Advice: this chapter is foundation of econometrics.
More informationECONOMETRIC THEORY. MODULE XVII Lecture - 43 Simultaneous Equations Models
ECONOMETRIC THEORY MODULE XVII Lecture - 43 Simultaneous Equations Models Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur 2 Estimation of parameters To estimate
More information2. Multivariate ARMA
2. Multivariate ARMA JEM 140: Quantitative Multivariate Finance IES, Charles University, Prague Summer 2018 JEM 140 () 2. Multivariate ARMA Summer 2018 1 / 19 Multivariate AR I Let r t = (r 1t,..., r kt
More informationLesson 14: Model Checking
Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@univaq.it Model checking Given the time series {x t ; t = 1,..., T } suppose that we have estimated
More informationOutline. Overview of Issues. Spatial Regression. Luc Anselin
Spatial Regression Luc Anselin University of Illinois, Urbana-Champaign http://www.spacestat.com Outline Overview of Issues Spatial Regression Specifications Space-Time Models Spatial Latent Variable Models
More informationFinancial Econometrics Prof. Massimo Guidolin
CLEFIN A.A. 2010/2011 Financial Econometrics Prof. Massimo Guidolin A Quick Review of Basic Estimation Metods 1. Were te OLS World Ends... Consider two time series 1: = { 1 2 } and 1: = { 1 2 }. At tis
More informationClass 4: VAR. Macroeconometrics - Fall October 11, Jacek Suda, Banque de France
VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Jacek Suda, Banque de France October 11, 2013 VAR IRF Short-run Restrictions Long-run Restrictions Granger Summary Outline Outline:
More informationMultivariate Time Series: VAR(p) Processes and Models
Multivariate Time Series: VAR(p) Processes and Models A VAR(p) model, for p > 0 is X t = φ 0 + Φ 1 X t 1 + + Φ p X t p + A t, where X t, φ 0, and X t i are k-vectors, Φ 1,..., Φ p are k k matrices, with
More informationSpatial Regression. 13. Spatial Panels (1) Luc Anselin. Copyright 2017 by Luc Anselin, All Rights Reserved
Spatial Regression 13. Spatial Panels (1) Luc Anselin http://spatial.uchicago.edu 1 basic concepts dynamic panels pooled spatial panels 2 Basic Concepts 3 Data Structures 4 Two-Dimensional Data cross-section/space
More informationLesson 13: Box-Jenkins Modeling Strategy for building ARMA models
Lesson 13: Box-Jenkins Modeling Strategy for building ARMA models Facoltà di Economia Università dell Aquila umberto.triacca@gmail.com Introduction In this lesson we present a method to construct an ARMA(p,
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 2 Jakub Mućk Econometrics of Panel Data Meeting # 2 1 / 26 Outline 1 Fixed effects model The Least Squares Dummy Variable Estimator The Fixed Effect (Within
More informationThe Multiple Regression Model Estimation
Lesson 5 The Multiple Regression Model Estimation Pilar González and Susan Orbe Dpt Applied Econometrics III (Econometrics and Statistics) Pilar González and Susan Orbe OCW 2014 Lesson 5 Regression model:
More informationBlock Bootstrap Prediction Intervals for Vector Autoregression
Department of Economics Working Paper Block Bootstrap Prediction Intervals for Vector Autoregression Jing Li Miami University 2013 Working Paper # - 2013-04 Block Bootstrap Prediction Intervals for Vector
More informationSTA 6857 VAR, VARMA, VARMAX ( 5.7)
STA 6857 VAR, VARMA, VARMAX ( 5.7) Outline 1 Multivariate Time Series Modeling 2 VAR 3 VARIMA/VARMAX Arthur Berg STA 6857 VAR, VARMA, VARMAX ( 5.7) 2/ 16 Outline 1 Multivariate Time Series Modeling 2 VAR
More informationExogeneity and Causality
Università di Pavia Exogeneity and Causality Eduardo Rossi University of Pavia Factorization of the density DGP: D t (x t χ t 1, d t ; Ψ) x t represent all the variables in the economy. The econometric
More informationNew Introduction to Multiple Time Series Analysis
Helmut Lütkepohl New Introduction to Multiple Time Series Analysis With 49 Figures and 36 Tables Springer Contents 1 Introduction 1 1.1 Objectives of Analyzing Multiple Time Series 1 1.2 Some Basics 2
More informationEconometrics I Lecture 3: The Simple Linear Regression Model
Econometrics I Lecture 3: The Simple Linear Regression Model Mohammad Vesal Graduate School of Management and Economics Sharif University of Technology 44716 Fall 1397 1 / 32 Outline Introduction Estimating
More informationInstrumental Variables, Simultaneous and Systems of Equations
Chapter 6 Instrumental Variables, Simultaneous and Systems of Equations 61 Instrumental variables In the linear regression model y i = x iβ + ε i (61) we have been assuming that bf x i and ε i are uncorrelated
More informationComputational Macroeconomics. Prof. Dr. Maik Wolters Friedrich Schiller University Jena
Computational Macroeconomics Prof. Dr. Maik Wolters Friedrich Schiller University Jena Overview Objective: Learn doing empirical and applied theoretical work in monetary macroeconomics Implementing macroeconomic
More informationSimple Linear Regression Model & Introduction to. OLS Estimation
Inside ECOOMICS Introduction to Econometrics Simple Linear Regression Model & Introduction to Introduction OLS Estimation We are interested in a model that explains a variable y in terms of other variables
More informationQuestions and Answers on Unit Roots, Cointegration, VARs and VECMs
Questions and Answers on Unit Roots, Cointegration, VARs and VECMs L. Magee Winter, 2012 1. Let ɛ t, t = 1,..., T be a series of independent draws from a N[0,1] distribution. Let w t, t = 1,..., T, be
More informationUnit roots in vector time series. Scalar autoregression True model: y t 1 y t1 2 y t2 p y tp t Estimated model: y t c y t1 1 y t1 2 y t2
Unit roots in vector time series A. Vector autoregressions with unit roots Scalar autoregression True model: y t y t y t p y tp t Estimated model: y t c y t y t y t p y tp t Results: T j j is asymptotically
More informationOil price and macroeconomy in Russia. Abstract
Oil price and macroeconomy in Russia Katsuya Ito Fukuoka University Abstract In this note, using the VEC model we attempt to empirically investigate the effects of oil price and monetary shocks on the
More informationQuantitative Analysis of Financial Markets. Summary of Part II. Key Concepts & Formulas. Christopher Ting. November 11, 2017
Summary of Part II Key Concepts & Formulas Christopher Ting November 11, 2017 christopherting@smu.edu.sg http://www.mysmu.edu/faculty/christophert/ Christopher Ting 1 of 16 Why Regression Analysis? Understand
More informationProperties of the least squares estimates
Properties of the least squares estimates 2019-01-18 Warmup Let a and b be scalar constants, and X be a scalar random variable. Fill in the blanks E ax + b) = Var ax + b) = Goal Recall that the least squares
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY PREFACE xiii 1 Difference Equations 1.1. First-Order Difference Equations 1 1.2. pth-order Difference Equations 7
More informationUnit Root and Cointegration
Unit Root and Cointegration Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt@illinois.edu Oct 7th, 016 C. Hurtado (UIUC - Economics) Applied Econometrics On the
More informationWeighted Least Squares
Weighted Least Squares The standard linear model assumes that Var(ε i ) = σ 2 for i = 1,..., n. As we have seen, however, there are instances where Var(Y X = x i ) = Var(ε i ) = σ2 w i. Here w 1,..., w
More information3. Linear Regression With a Single Regressor
3. Linear Regression With a Single Regressor Econometrics: (I) Application of statistical methods in empirical research Testing economic theory with real-world data (data analysis) 56 Econometrics: (II)
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY & Contents PREFACE xiii 1 1.1. 1.2. Difference Equations First-Order Difference Equations 1 /?th-order Difference
More information10. Time series regression and forecasting
10. Time series regression and forecasting Key feature of this section: Analysis of data on a single entity observed at multiple points in time (time series data) Typical research questions: What is the
More informationEcon 583 Final Exam Fall 2008
Econ 583 Final Exam Fall 2008 Eric Zivot December 11, 2008 Exam is due at 9:00 am in my office on Friday, December 12. 1 Maximum Likelihood Estimation and Asymptotic Theory Let X 1,...,X n be iid random
More informationEmpirical Economic Research, Part II
Based on the text book by Ramanathan: Introductory Econometrics Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna December 7, 2011 Outline Introduction
More informationIntroduction to Econometrics Midterm Examination Fall 2005 Answer Key
Introduction to Econometrics Midterm Examination Fall 2005 Answer Key Please answer all of the questions and show your work Clearly indicate your final answer to each question If you think a question is
More informationEmpirical Market Microstructure Analysis (EMMA)
Empirical Market Microstructure Analysis (EMMA) Lecture 3: Statistical Building Blocks and Econometric Basics Prof. Dr. Michael Stein michael.stein@vwl.uni-freiburg.de Albert-Ludwigs-University of Freiburg
More information